Questions
 Chepkemoi bought a new washing machine for Kshs.420,000. Its value depreciated over
the next 5years at the following rates; 15%, 13%, 12%, 9% and 7%. For the next 6 years,
the rate of depreciation remained constant at 5% then the rate of depreciation remained at 4%
each. How long did it take for the value of the washing machine to be 1/3 of its original value?  The table below shows income tax rates for the year 2006
Taxable income in shs. Pa Rate of tax in % 1 – 120,000 10 120,001 – 240,000 15 240,001 – 360,000 25 360,001 – 480,000 35 Over 480,000 50  The figure below represents two pulley wheels, centres A and B with a rubber band CDEFGHC stretched round them.
Radius of wheel centre A = 16cm, AB = 30cm. CD, GF are tangents to the circles ∠ CAB = 86.3° calculate the length of the belt CD
 Find the angle ABD
 Find the length of the belt that would go round the pulleys (CDFGHC)
 In the figure below, ABCD is a cyclic quadrilateral and BD is a diagonal. EADF is a straight line, ∠CDF = 68^{o}, ∠BDC = 45^{o }and ∠BAE = 98^{o}.
Calculate the size of: ∠ABD.
 ∠CBD
 A customer deposited Ksh.15,500 in a savings account. Find the accumulated amount after 3½ years if interest was paid at 16% per annum compounded semiannually
 A retailer mixes three types of rice, Bismatti costing shs.120 per tin with Pishori costing shs.150 per tin and Ahero rice costing shs.80 per tin in the ratio x : 1 : 2 respectively. If he sells the mixture at shs.137.50 per tin making a profit of 25%. Calculate the value of x.
 Ashanti is a saleswoman and earns a commission on sales based on the monthly rates shown in the table below:
Sales (Kshs) Commission rate % of sales The first 5000 10% The next 3000 15% Sales above 8000 20%  The table below shows the annual income tax rates for a certain year.
Total income per month in Kshs. Rates in Kshs. Per £ 110164
10165 – 19740
19741 – 29316
29317 – 33892
388983 and above2
3
4
5
6Automatic personal relief shs.1162
Calculate: His taxable income per month
 His monthly tax payable
 His annual tax payable
 A company employee earns a basic salary of Kshs.25,000 and is also given taxable allowances amounting to Kshs.10,480. Using the table of taxation above:
 Calculate the employee’s taxable income
 If the employee is entitled to a personal tax relief of Kshs.800 per month, determine the net tax
 If the employee was given 40% increase in his income, calculate the percentage increase in his income tax
 A certain amount of money was invested at compound interest of 10% compounded every two years for ten years. Given that the investor invested a total of 500,000/= at the end of the ten years, find the amount of money invested to the nearest shillings
 The cash price of a T.V set is Ksh. 26,000. Linda bought the set on hire purchase terms by paying a deposit of Ksh. 6,000 and the balance by 24 equal monthly installments of Kshs. 1,045.30. Find the compound rate interest per year.
 What would Kshs. 15000 amount to after 3 years at 16% per annum compounded quartely?
 Income rates for income earned were charged as follows:
Income in Kshs. p.m. Rate in Kshs per sh.20 1  8400
8,401  18,000
18,001  30,000
30,001  36,000
36,001  48,000
48,001 and above2
3
4
5
6
7
He is entitled to a family relief of Kshs. 1240 per month.
Determine: His taxable income per month.
 Net tax.
 In addition, the following deductions were made
NHIF shs. 230
Service charge Kshs. 100
Loan repayment Kshs. 4000
Cooperative shares of Kshs. 1200
Calculate his net salary per month
 Use the taxation rates in the table below to answer the questions that follow;
Taxable income in K £ p.a Rate % per K£ 14500
45017500
7501 – 10500
10501 – 13500
13501 – 16500
0ver 1650010
15
20
25
30
35 The managers total deductions per month
 Total tax per month
 The manager’s annual gross salary
 The manager’s monthly basic salary if her monthly allowance and medical allowances are 10000 and 2000 shillings
 The table below shows the income tax for a certain year;
Monthly taxable income (Kshs.) Tax rates (%) 1 9680
9681 18800
18801 – 27920
27921 – 37040
37940 and above10%
15%
20%
25%
30%
N.H.I.F Kshs.320
Union dues Kshs.200
Cooperative shares Kshs.7,500 Calculate Odero’s monthly basic salary in Kshs
 Calculate his monthly salary
 A car is worth shs.800,000 when new. During the first year it depreciates by 20% of its value and in the second it deprecates by 5% of its value at the start of the year. During the third, fourth and fifth year, depreciation rate is 10%. How much less will it cost at the end of the fifth year?
 Find by how much the compound interest will exceed simple interest on shs.3,000 for two years at 15% per year
 The table below shows the income tax rates:
Income per month (K£) Rate in Kshs per £
1  325 2
326 – 975 3
976  1300 5
1301 – 1625 6
Over 1625 7.50
Mr. Misoi is a public servant who lives in a government house and pays a nominal rent of Kshs.1,220 per month. He earns a basic salary of Kshs. 24,800 and a house allowance of Kshs.12,000 per month. He is entitled to a monthly relief of kshs.1620. Calculate his monthly;
 Taxable income in K£
 Tax payable without relief
 Tax after relief
 Apart from the income tax. The following monthly deductions are made from his earnings
HELB loan repayment Kshs.2400
 Health insurance fund Kshs.1200
 2% of Basic salary union fee
Calculate: the total monthly deduction made on Mr. Misoi’s income
 Mr. Misoi’s net income per month
 Calculate his monthly;
 Joseph bought a camera on hire purchase (H.P) term by paying a deposit of shs.7200 and cleared the balance in 24 equal monthly installments each of 1250.
 find the hire purchase price of the camera
 the hire purchase price of the camera is 24% higher than the cash price. Find the cash price of the camera
 Kangara took a loan from a financial institution and bought the camera with cash. He repaid the loan at 18% p.a compound interest at the end of the two years. Find the total interest paid by Kangara.
 Income tax for all the income earned was charged at the rates shown.
Total Income p.a (K.£) Rate in sh per K£ 1 – 1980
1981 – 3960
3961 – 6440
6441 – 7920
7921 – 9900
Excess of 99002
3
5
7
9
10 Wanyonyi earned a salary of Kshs.10,500 per month. In addition he was given a house allowance of Kshs. 6500 per month. He got tax relief of Kshs. 300 per month.
Find ; His taxable income p.a
 Income tax he pays per month.
 A part from income tax the following deductions are made per month. NHIF of Kshs
 Wanyonyi earned a salary of Kshs.10,500 per month. In addition he was given a house allowance of Kshs. 6500 per month. He got tax relief of Kshs. 300 per month.
Answers
 After 1st year = (^{95}/_{100} x 4200000)
= Shs.357,000
After 2nd year = ( ^{87}/_{100} x 357000)
= sh310590
After 3rd year = (^{88}/_{100} x 310590
= shs.273319.20
After 4th year = ( ^{91}/_{100} x 273319.20)
=shs.248720.50
After 5th year = (248720.50 x ^{93}/_{100})
The next 6 years = shs. 231310
A = 231310 (1 0.05)6 = 170034.10
Then 140000 = 170034.10 (10.04)^{n}(0.96)^{n} = ^{140000}/_{170034.10} = 0.8234
n = ^{log 0.8234}/_{log 0.96}= ^{0.0844}/_{0.01773} = 4.76yrs
Total no. of years = 5 + 6 + 4.76yrs = 15.76years  Gross tax = 4830 + 1120 + 600 = sh 6550 per month
Annual gross tax = 6550 X 12= 78,600
10 X 120,000 = sh.12,000
100
15 X 120,000 = sh.18,000
100
25 X 120,000 = sh. 30,000
100
Re. tax = 78600 – (12000 + 18000+30000)
=78600 – 60,000 = 18,6000
35 X x =18,600
100
0.35x = 18,600
x = sh 53142.86
Taxable income p.a = 36,000+53142.86 =sh.412142.86
Monthly salary = ^{413142.86}/_{12} + 12,000
= 34428.57 + 1200 = Sh 35628.57 
 Sin 86.3° = ^{XB}/_{AB}Sin 86.3°= ^{XB}/_{30}XB = 30sin86.3°
XB = CD = 29.93746855 cm  ∠ ABX = 90°  86.3°
= 3.7°
∴∠ ABD = 3.7° + 90°
= 93.7°  ∠ DBF obtuse = 360°  187.4°
= 172.6°
Arc DEF = ^{ø}/_{360} πD or ^{ø}/_{360 }x2πr
But cos 86.3° = ^{AX}/_{AB}Cos 86.3° = ^{AX}/_{30}AX = 1.935969248 cm
DB = 16 – 1.935969248 = 14.06403075 cm
∴ Arc DEF = ^{172.6°}/_{360° }x ^{22 }/_{7 }x 14.06403075
= 106807.8751
2520
= 42.38407742 cm
Arc CGH
∠reflex CAG = 360°  (2x 86.3°)
= 187.4°
∴Arc CGH =187.4°/ 360° x2 x22 /7 x 16
= 131,929.6
2520
= 52.35301587 cm
Total length of belt to go round the belt
= CD + DEF + GF + CHG
= 29.93746855 + 42.38407742 + 29.93746855 + 52.35301587
= 154.6120304 cm
 Sin 86.3° = ^{XB}/_{AB}Sin 86.3°= ^{XB}/_{30}XB = 30sin86.3°
 ∠ABD = 31^{0}∠ CBD = 37^{0}
 A = 15,500(1 + ^{8}/_{100})^{7}
= Ksh.25,707 
 Commission earned Kshs. (8368 – 6700) = Kshs. 1668/=
let sales in 3rd bracket be y
^{10}/_{100 }x 5000) + (^{15}/_{100 }x 3000) + (^{20}/_{100 }x y) = 1668
500 + 450 + 0.2y = 1668
0.2y = 1668 – 950 = 718
y =^{718}/_{0.2 }= 35%
Total sales = (8000 + 3590)
= shs.11590 
 Taxable income = (25000 + 12000 + 3000)= 40000
 Income tax
10164 x ^{2}/_{20 }= Shs.1016.40
10164 x ^{3}/_{20} = Shs.1524.60
10164 x ^{4}/_{20 }= Shs.2032.80
Remaining :
9508 x ^{5}/_{20 }= Shs.2377
Total tax payable p.m = 6950.8 – 1162 = Shs.5788.80  Annual tax payable = 5788.80 x 12 = Shs.69465.60

 taxable income = Kshs. 25000 + Kshs.10480
= Kshs. 35480  tax charged:
1st 4350 = 4350 x ^{2}/_{20 }= 683.25
2nd 4555 = 4555 x ^{3}/_{20 }– 683.25
3rd 4555 = 4555 x ^{4}/_{20 }– 911
4th 4555 = 4555 x ^{5}/_{20 } 1138.75
Rem. 17465 = 17645 x ^{6}/_{20} – 5239
Total tax = 8407.5  800.00 = 7607.50  ^{40}/_{100} x 35480 – 14.192 = 49672
New income = 35480 + 14192 = 49672
Remainder = 49672 – 18015 = 31657
Tax charged = 31657 x 6/20 =12665.1
Total tax = 12665.1
% increase in income ax = 4257.6 x 100
7607.5 = 55.97%
 taxable income = Kshs. 25000 + Kshs.10480
 A = P(^{HR}/_{100})^{n}500000 = P( 1+ ^{20}/_{100})^{5}500,000 = (^{120}/_{100})^{5}500,000 = P
(1.2)^{5}P = Shs.200,938.786 ≈shs. 200,939  Principal = 26, 000  6,000 = 20,000
Total H.P instalments = 1045.3 x 24= 25087.20
25087.20 = 20,000 [^{1}/_{100} + r]^{2}1.254 = [1+ ^{r}/_{100}]^{2}1.120 = ^{1}/_{100} + r
^{r}/_{100} = 0.12 or 12%  No. of periods =12
r = 4% per period
A = 1.0412 x 15000
= 24015.5 
 Taxable income = 19200 + 12000 + 1300 + 2300 = 34800
 Net tax
8400 x ^{2}/_{20 }= 840
9600 x ^{3}/_{20} = 1440
12000 x ^{4}/_{20 }= 2400
4800 x ^{5}/_{20} = 1200/5800
Net tax = 5800 – 1240
= 4560  Net salary = 34800 – (4560 + 5530)
= 24710

 9000 + 350 + 800 + 1200 = 11350
 9000 + 3000 = 12000
 Total taxes = 12000 x 12
= shs.144000p.a
Taxes
450 x 2 = shs.9000
3000 x 3 = shs.9000
3000 x 4 = shs.12000
3000 x 5 = shs.15000
3000 x 6 = shs.18000
Shs.63,000
144000 – 63000= shs.81000
7y = 81000y = 11571
Taxable income= 4500 + 3000 x 4 + 11571 = K 28071p.a
Gross salary = shs. 561420p.a  Total allowances = 12000 x 12
= 144,000
Basic salary = 561420  144000
= Shs.417,420
Monthly basic pay = shs. 34785

 Net tax 5512
Add relief 1162
Tax payable 6674
Tax on 9680 earned
9680 X ^{10}/_{100 }= 968
Tax on 9120 earned
9120 x ^{15}/_{100 }= Shs. 1368
Tax on next 9120 x ^{20}/_{100 }= Shs.1824
Tax on next 9120 x ^{25}/_{100 }= 2280
Total 968 + 1368 + 1824 + 2280 = 6440
6674 – 6440 = 234
Let x be charged at 30%
^{30}/_{100 }X x = 234
X = ^{(234 X 100)}/_{30} = Shs.780
Total chargeable Income
780 + (9120 x 3) + 9680 = 37820
Salary 37820 – 15220 = Shs.2260 per month.  Net salary (37820 – 1270 – 6674) = Shs.29876
 Net tax 5512

 1st year after dep. Of 20%
800 000 x ^{80}/_{100}= Khs. 640,000……………………………………………………
2nd year after dep. of 5%
= 640000 x ^{95}/_{100}= 608,000 .....................................
The next 3 yrs
A = P [1 – ^{R}/_{100}] = 608,000 ( 1 – ^{10}/_{100})^{3}= 698 000 (0.9)
= Sh. 443,232 ……………………………………………………
800,000 – 443,232 = Sh. 356,768 ………………………………………  S.I = 3000 x ^{15}/_{100 }x 2
= Sh 900 …………………………………………………
A = 3000 [1 + ^{15}/_{100}]^{2}= 3000[1.15]^{2}= sh. 3967.50 ……………………………………………….
C.I = sh 967.50
967.50 – 900 = sh 67.50 ……………………………………….
 1st year after dep. Of 20%

 Taxable Income
[^{115}/_{100} x 24 800[ + 12000  1220
= 28520 + 12000 – 1220
= Ksh.39,300
= K₤ 1965 p.m.  Tax due 325 x 2 = sh 650
650 x 3 = sh 1950
325 x 5 = sh 1725 ………………………..
325 x 6 = sh 1950
340 x 7.50 = sh 2250
Total tax = sh. 8825 P.m………………………(without relief)
 Taxable Income
 Total deduction
= sh (7280 + 2400 + 1200 + ^{2}/_{100} of 24 800)……………….
= (7280 + 2400 + 1200 + 496) + 1220
= sh (11376 +_ 1220) = sh. 12,596 P.m …………………………………………  Net income = sh (24800 + 1200 – 12596) = sh. 24,204 P.m……………………………
 Total deduction

 Total instalments = (24 x 1250)= Shs.30000
H.P = 7200 + 30000= 37200  124% = 37200
100% =
C.P = ^{100}/_{124} x 37200
= 30000  A = 30000 (1 + 18/100)2
= 30000 (1.18)2 = 41772
Total interest = 41772 – 30000 = 11772
 Total instalments = (24 x 1250)= Shs.30000


 (10, 500 + 6,500) x 12 = K₤ 10,20 p.a
20  1st 1980 x 2 = Kshs. 3960
2nd 1980 x 3 = Kshs 5940
3rd 2480 x 5 = Kshs. 12 400
4th 1480 x 7 = Kshs. 10360
5th 1980 x 9 = Kshs.17 820
Last 300 x 10 = Kshs 3 000
Kshs. 53 480
PAYE = ^{53480}/_{12}  3000 x ^{12}/_{100}= Shs. 4156.70  Net monthly pay
17000 – 320 +^{ 2}/_{100} x 17000
= 17000 – 660
= Kshs 16 340.0
 (10, 500 + 6,500) x 12 = K₤ 10,20 p.a

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